Visit http://www.mathsmadeeasy.co.uk/ for more fantastic resources. AQA, Edexcel, OCR A Level A Level Mathematics Understand and use double angle formulae Name: Total Marks: Maths Made Easy Complete Tuition Ltd 2017
C5- Understand and use double angle formulae; use of formulae for i ±, c ±, a ± ; understand geometrical proofs of these formulae- Answers AQA, Edexcel, OCR 1) For the following questions, and δ are all acute angles. Sin = cos( = tan(δ = The answers for the following questions are applications of the following formula. sin(a ± B) = sinacosb ± sinbcosa cos(a ± B) = cosacosb sinasinb tan(a ± B) = tan ±tan tantan (1) (2) (3) You also need to recall = ; sec = ; the double angle formulas are just si x csx applications of (1), (2) and (3), where B is replaced by another A. [1 mark for each correct answer- 8 max] Find exact values for: (a) sin( + (b) sin( (c) cos( + (d) cos( + δ + (e) cos( - δ (f) tan( - (g) tan( + δ (h) tan( + δ + + + + [1 mark for each correct answer- 8 max] Find exact values for: i sin j cos k tan l sin m cos n tan o secδ p cosecδ
2) Demonstrate geometric proof of the double angle formula for sine and cosine. For sine, we know the double angle formula is [1 mark for drawing] sin + = + We can demonstrate this geometrically by stacking two right-angle triangles on top of each other. The two triangles Triangle ACD, where is length 1 and Triangle ABC are shown below. There is also a triangle AFD and the side FD is oppposite angles x and y. If we A x y establish the length of FD we can prove the formula. At the moment we can say sin + = Writing DF as DE and EF gives sin + = + and writing EF as CB (same length as BCEF) is rectangle. sin + = + Establish some of the unknown lengths of the sides of the polygon. sin = h = = sin And similarly, we know that AC can be written as E D F We can now use this to establish length CB cos = h = = sin = h = cos = cos sin Angle BCE we know is the same size as angle y (CE is parallel to AB alternate angles). Therefore we know C B = 9 = 9 = as the angles in the triangle CDE must add up to 180. cos = h = = sin (1) (2) (3)
Inserting (2) and (3) into (1) gives = sin cos sin + = sin cos + cos sin For cosine, we know the double angle formula is cos + = cos cos sin sin. A x y E D F C B cos + = h h = cos + = cos + = = Using equation number (2) we can establish that cos = h = = cos = cos cos (4) (5) To obtain the length of FB, we can obtain the length of EC, and they are the same because BCEF is a rectangle. which we previously showed. Putting (5) and (6) into (4) gives = = sin sin = h = sin = sin sin = cos + = cos cos sin sin (6) 3) State the formula for i +, c + and use these to write the formula for a +. sin(a + B) = sinacosb + sinbcosa cos(a +B) = cosacosb - sinasinb We know that tan = six and that the same relationship is true for the double csx angle/additional formula.
Thus, we can write sin + tan + = cos + tan + = + If we divide each term by we get the following tan + = + Cancelling out the left- hand part of the numerator allows us to write it as tan + = + Cancelling out the right- hand part of the numerator allows us to write it as tan + = + Cancelling out the left-hand part of the numerator allows us to write it as tan + = + Now we have many terms with a numerator and denominator, meaning we can replace them with. tan + = +
4) Demonstrate using your knowledge of trigonometric identities that the following is true: We know that so we can write that We also know that Rearranging this gives Inserting (2) into (1) gives 5) Show cx = c x cx cs = si cos(a +B) = cosacosb sinasinb cos = cos = cos sin = sin + cos cos = sin cos = sin sin cos = sin cos = cos + = cos cos sin sin Inserting double angle formulas for cosine and sine gives Replacing sin with cos cos = cos sin sin cos = cos cos cos cos = cos + cos + cos cos = cos (1) (2)
6) Simplify the following csx six + csx Using the double angle formula for cos allows us to write cos sin sin + cos Spotting that the numerator is the difference of two squares, means we can rewrite it as cos sincos + sin sin + cos And as the bracket on the right of the numerator is the same as the denominator they will cancel to give 1, leaving us with: cos sin